The person who installed the browser extension would be paying the subscription and ignoring said clause.
Also, extension likely has also terms of usage prohibiting uploading copyrighted content shifting liability on users.
They went after seeders
1: https://chrome.google.com/webstore/detail/archive-page/gcaim...
I think the distinction between the two is moot in this post. The question could very well have been "How does archive.org bypass paywalls?" Though it's interesting that archive.is seems to just crawl the URL. Indeed that means they wouldn't necessarily be able to bypass the paywall.